GG274 Homework 8: Hypothesis Testing

GG274 Homework 8: Hypothesis Testing#

Logistics#

Due date: The homework is due 23:59 on Monday, March 10.

You will submit your work on MarkUs. To submit your work:

  1. Download this file (Homework_8.ipynb) from JupyterHub. (See our JupyterHub Guide for detailed instructions.)

  2. Submit this file to MarkUs under the hw8 assignment. (See our MarkUs Guide for detailed instructions.) All homeworks will take place in a Jupyter notebook (like this one). When you are done, you will download this notebook and submit it to MarkUs.

Introduction#

Residential instability is one component of the Ontario Marginalization Index that includes indicators of types and density of residential accommodations, and certain family structure characteristics, such as living alone and dwelling owndership. (see OCHPP)

In this homework you will explore the following question:

Are mental health visits different in Toronto neighbourhoods with higher “residential instability”?

import numpy as np
import pandas as pd
%pip install xlrd

Step 1 - Read the Neighbourhood Instability data into a pandas DataFrame#

a) The data is stored in 1_marg_neighb_toronto_2006_OnMarg.xls - a Microsoft Excel file format with file extension .xls.

Use the pandas function read_excel to read the sheet Neighbourhood_Toronto_OnMarg into a pandas DataFrame named marg_neighbourhoods.

# Write your code below
marg_neighbourhoods

Use marg_neighbourhoods to create a another DataFrame called instability_df that has three columns: "Neighb id ", "Neighbourhood name ", "INSTABILITY".

# Write your code below
instability_df

# Check the columns
list(instability_df)

b) Rename the column names of instability_df using the following table. The DataFrame with the new column names should be called instability_df (i.e., don’t change the name of the DataFrame).

Original column name

New column name

“Neighb id “

n_id

“INSTABILITY”

instability

“Neighbourhood name “

name

Be aware of the trailing spaces!

# Write your code below

Step 2 - Read the mental health visit data into a pandas DataFrame.#

a) In this step you will read in data on rates of mental health visits stored in 2_ahd_neighb_db_ast_hbp_mhv_copd_2012.xls into a pandas DataFrame named mental_health_neighbourhoods.

# Write your code below
mental_health_neighbourhoods

b) Create a new DataFrame mh_visit_rates by selecting the columns in mental_health_neighbourhoods that corresponds to Neighbourhood ID, Neighbourhood Name, and ‘Age-Standardized rate of Mental Health Visits (2012), All Ages 20+’ rename this column in mh_visit_rates to mh_visit_rates_mf. When you rename this column don’t change the name of the DataFrame mh_visit_rates.

# Write your code below
mh_visit_rates

Step 3 - Merge mental health visits and instability#

In this step you will merge the mh_visit_rates with mental_health_neighbourhoods.

a) Merge mh_visit_rates with instability_df and name this DataFrame mh_visit_instability.

# Write your code below
mh_visit_instability

Step 4#

a) Create a new column in mh_visit_instability named instability_binary that categorizes neighbourhoods. The new columns should have two possible values:

  • "High", if the neighbourhood’s instability value is greater than or equal to the mean of "instability"

  • "Low", if the neighbourhood’s instability value is less than the mean of "instability"

# Write your code below

b) Compute the frequency distribution of instability_binary. Save the results in instability_binary_frequencies.

# Write your code below
instability_binary_frequencies

c) Is there evidence that Toronto has many neighbourhoods that have residential instability? Briefly explain. (1 mark)

Answer here

Step 5 - Do neighbourhoods with high residential instability have more mental health visits compared to neighbourhoods with low residential isntability?#

a) Use the DataFrame describe method to compute the distribution of mh_visit_rates_mf in mh_visit_instability grouped by instability_binary. Store the results in median_table.

# Write your code below
median_table

Use median_table to compute the difference in medians between neighbourhoods with high and low instability. Store this value in median_diff.

# Write your code below
median_diff

Step 6 - Set up a simulation in python to test if the medians are equal#

a) In this step you will write a function random_shuffle_median that returns a simulated value of the median difference (a simulated value of the test statistic) of mental health visit rates in neighbourhoods with high versus low residential instability assuming that there really is no difference in mental health visit rates between these types of neighbourhoods.

A step-by-step explantion of a similar function was given in lecture, and you can follow this example to help guide you through this step.

The function random_shuffle_median is started for you below. Your task is to complete the function by filling in the ....

Try writing a meaningful docstring for random_shuffle_median. The pandas docstring guide has some great examples and guidelines. (NB: this will not be graded)

def random_shuffle_median():
    """
    Put your docstring here (optional)
    """

    # shuffle the column of mh_visit_instability that corresponds to high/low instability

    instability_binary_shuffle = mh_visit_instability[...].sample(
        frac=1, replace=False).reset_index(drop=True)
    
    # calculate the median visit rate for high and low instability neighbourhoods

    visitrate_low_shuffle = mh_visit_instability.loc[instability_binary_shuffle == ..., ...].median()
    
    visitrate_high_shuffle  = mh_visit_instability.loc[instability_binary_shuffle ==  ..., ...].median()
    
    shuffled_diff = ... - ...
    
    return shuffled_diff

print(random_shuffle_median.__doc__)

b) Explain the purpose of

mh_visit_instability[...].sample(frac=1, replace=False).reset_index(drop=True)

in 1-2 sentences.

Answer here

Step 7 - Compute the distribution of simulated values of the median difference assuming the null hypothesis is true#

We will use your student number to generate data for this homework. Complete the assignment statement below by typing your student number as an int. In other words assign your student number as an integer to the variable student_number.

# Replace the ... with your student number
student_number = ...

# This checks that you correctly typed in your student_number as an int.
# Make sure there's no error when you run this cell!
assert type(student_number) == int

a) Write a function called shuffled_diffs that returns a list. The function should use a for loop that iterates the function random_shuffle_median an arbitrary number of times. The number of times that the for loop iterates should be controlled by a function parameter named number_of_shuffles.

# Write your code below

b) Use shuffled_diffs to compute 10 000 simulated median differences between high and low instability neighbourhoods assuming that there is no difference in median mental health visit rates between high and low instability neighbourhoods. Store the values in shuffled_diffs_10000.

np.random.seed(student_number)

# Write your code below

c) Plot the distribution of the 10,000 simulated values stored in shuffled_diffs_10000 using a matplotlib histogram. Name the plot null_hypothesis_distribution_plot. Label the horizontal axis as "Difference in median visit rates for high and low instability neighbourhoods" and the vertical axis as "Frequency".

import matplotlib.pyplot as plt

# Write your code below

Step 8 - Compute the p-value#

a) Compute the number of simulated differences in medians in shuffled_diffs_10000 that are greater than or equal to the observed median difference (median_diff). Store this value in right_extreme.

# Write your code below

b) Compute the number of simulated differences in medians in shuffled_diffs_10000 that are less than the observed median difference (median_diff). Store this value in left_extreme.

# Write your code below

c) Use right_extreme and left_extreme to compute the p-value. Store the p-value in pvalue.

# Write your code below

Step 9 - Communicate what you did in the steps above#

a) In a few sentences introduce the question that you explored in this homework (see the beginning of this homework). For example, why do you or others think this is an important question? (1 mark)

answer goes here …

b) Briefly describe the data sources that you used to answer the question. Which statistical variables did you use and why did you use these varaibles? (1 mark)

answer goes here …

c) What computational and statistical methods or analyses did you use to answer the question? Briefly describe these methods and how they were used to answer the question. (1 mark)

answer goes here …

d) Briefly describe the results of your statistical analysis in a few sentences. (1 mark)

answer goes here …

e) What conclusions can you draw about the question you set out to answer that is supported by the data and statistical analysis of the data? State at least one limitation of your conclusions. (See the USC Research Guide section on study limitations) (1 mark)

answer goes here …

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